Now, assuming that the vibrato is unremovable/unreducable, what should the recording engineering do? Should ask the player to sing without vibrato, or what?
I'll assume that this is a serious question, and treat it as such. Yes, if no vibrato is the goal, then it should really be sung and recorded without vibrato. And for the record, I usually don't have any problems taming both pitch mod. and pitch drift if needed.
The reason I say that it's difficult (at times) is because sometimes it's a combination of volume, pitch and timbre changes that produce the "vibrato". And to remove all those factors from the performance, while keeping the sound natural, can be hard to do. Depends on the singer really.
Like Claudio, I'd be interested in hearing an example.